Expanding and Condensing Logarithms

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or $e$, we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.

To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.

Given any positive real numbers $M,b$, and $n$, where $n \neq 1$ and $b \neq 1$, we show

$\log _{b} M=\frac{\log _{n} M}{\log _{n} b}$

Let $y=log_bM$. By exponentiating both sides with base $b$, we arrive at an exponential form, namely $b^y=M$. It follows that

$\begin{array}{lll} \log _{n}\left(b^{y}\right) & =\log _{n} M & \text { Apply the one-to-one property. } \\ y \log _{n} b & =\log _{n} M & \text { Apply the power rule for logarithms. } \\ y & =\frac{\log _{n} M}{\log _{n} b} & \text { Isolate } y . \\ \log _{b} M & =\frac{\log _{n} M}{\log _{n} b} & \text { Substitute for } y . \end{array}$

For example, to evaluate $log_536$ using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

$\begin{aligned} \log _{5} 36 &=\frac{\log (36)}{\log (5)} \quad \text { Apply the change of base formula using base } 10 \text {. }\\ &\approx 2.2266 \quad \text { Use a calculator to evaluate to } 4 \text { decimal places. } \end{aligned}$

THE CHANGE-OF-BASE FORMULA

The change-of-base formula can be used to evaluate a logarithm with any base.

For any positive real numbers $M,b$, and $n$, where $n \neq 1$ and $b \neq 1$,

$\log _{b} M=\frac{\log _{n} M}{\log _{n} b}$

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

$\log _{b} M=\frac{\ln M}{\ln b}$

and

$\log _{b} M=\frac{\log M}{\log b}$

HOW TO

Given a logarithm with the form $\log _{b} M$, use the change-of-base formula to rewrite it as a quotient of logs with any positive base $n$, where $n \neq 1$.

1. Determine the new base $n$, remembering that the common $\log , \log (x)$, has base 10 , and the natural $\log , \ln (x)$, has base $e$.
2. Rewrite the log as a quotient using the change-of-base formula
   1. The numerator of the quotient will be a logarithm with base $n$ and argument $M$.
   2. The denominator of the quotient will be a logarithm with base $n$ and argument $b$.

EXAMPLE 13

Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change $log_53$ to a quotient of natural logarithms.

Solution

Because we will be expressing $\log _{5} 3$ as a quotient of natural logarithms, the new base, $n=e$.

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

$\begin{aligned} \log _{b} M &=\frac{\ln M}{\ln b} \\ \log _{5} 3 &=\frac{\ln 3}{\ln 5} \end{aligned}$

TRY IT #13

Change $log_{0.5}8$ to a quotient of natural logarithms.

Q&A

Can we change common logarithms to natural logarithms?

Yes. Remember that $log9$ means $log_{10}9$. So, $log9= \frac{\ln9}{\ln10}$.

EXAMPLE 14

Using the Change-of-Base Formula with a Calculator

Evaluate $log_2(10)$ using the change-of-base formula with a calculator.

Solution

According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base $e$.

$\begin{aligned} &\log _{2} 10=\frac{\ln 10}{\ln 2} & & \text { Apply the change of base formula using base e }. \\ &\approx 3.3219 & & \text { Use a calculator to evaluate to 4 decimal places. } \end{aligned}$

TRY IT #14

Evaluate $log_5(100)$ using the change-of-base formula.